Dept Computer Science Korea Univ Intelligent Information System

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. A I (Artificial Intelligence) Professor I. J. Chung

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. A I (Artificial Intelligence) Statistical Reasoning Model to uncertainty S. R (D/S) F. L. CF certain factor CF[h, e] = MB[h, e] - MD[h, e] MB[h, e] Measure(let 0 and 1) of belief in hypothesis h given the evidence e. MB measures the extent to which the evidence supports the hypothesis if the evidence fails to support the hypothesis MB=0 MD[h, e] Measure(let 0 and 1) of disbelief in hypothesis h given the evidence e. MD measures the extent to which the evidence supports the negation of the hypothesis if the evidence supports the hypothesis, it is 0.

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. A I (Artificial Intelligence) Dempster / Shaffer theory(D/S theory of evidence) Statistical reasoning (probabilistic reasoning) Approach to uncertainly (alternative : f. l. ) Bayesian reasoning : conditional based on probability P(A)+P(~A) ≤ 1 Universe Ω → frame of discernment with n hypothesis, exactly 1 is true (mutually exclusive) : initial set of all hypothesis in the problem domain m : basic probability function on Ω m : 2Ω → [0, 1] subset A ⊆ U → [0, 1] m(x) ≥ 0 m( ) = 0 m(x) = 1 m(x) : measure of belief committed exactly to x

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. A I (Artificial Intelligence) m : impact of a piece of evidence on the confidence / belief. probability discernment on 2 u → all subsets of U : measure of belief committed to A. x⊆Ω, i. e. x∈2Ω : focal element in m if m(x) > 0. core of m, k(m) : set of all focal elements in m. Probability should be given to all subsets of U v. t. to ind. single members. Bel(x) : Belief function (credibility function) Bel : 2Ω → [0, 1] Bel(x) = m(y) for each x⊆Ω i. e. x∈2Ω Sum of belief committed to every subset of x by m. Measure of total support or belief committed to the set x. E. g. Bel({A, C, D})=m({A, C, D})+ … +m({D})+m( ) 0

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. A I (Artificial Intelligence) Properties 1) Bel(Ω)=1 (∵ m(y)=1) 2) For ∀x⊆Ω with exactly one element, Bel(x)=m(x) 3) Bel(x)+Bel(x) ≤ 1 ∵ Bel(Ω) = Bel(x∪x) = Bel(x)+ m(y)=1 4) Bel(x)+Bel(y) ≤ Bel(x∪y) for x, y ≤ Ω Plausibility function Pl : 2Ω →[0, 1], Pl(x)= m(y) for ∀x ⊆ Ω upper bound for belief in x Bel : lower bound for belief is x Pl(x) = 1 -Bel(x) Pl(x) : total confidence not assigned to x upper bound to the real confidence in x Pl(x)-Bel(x) : uncertainly with respect to x [Bel(x), Pl(x)] : belief interval of x (confidence in x)

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. A I (Artificial Intelligence) [Bel(x), Pl(x)] = [0, 1] : no information concerning x is available [Bel(x), Pl(x)] = [1, 1] : completely confirmed by m [Bel(x), Pl(x)] = [0, 0] : represent the belief that the hypothesis is absolutely false [Bel(x), Pl(x)] = [0. 3, 1] : ∃ some evidence in favor of the hypothesis x [Bel(x), Pl(x)] = [0. 15, 0. 75] : ∃ some evidence in favor as well as against x D/S : 2 pieces of evidence & new basic plausibility function describing the combined influence of the pieces of evidence → D/S rule of combination

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. A I (Artificial Intelligence) E. g. ∃ 4 possible disorders to be considered as a diagnosis Ω = { heart attack, pericarditis, pulmonary embolism, aortic dissection } H, P : heart disorder PE, A : blood vessel disorder if evidence pointing at a certain diagnosis in particular, the probability of entire frame of discernment is 1. ie, ∀ proper subset of Ω gets the basic probability #0. core of m 0={Ω} m 0(x)= 1 0 if x= Ω elsewhere ∃ evidence of heart disorder with 0. 4 m 1(x)= 0. 6 0. 4 if x= Ω if x= {heart attack, pericarditis} 0 elsewhere K = { heart attack, pericarditis}

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. A I (Artificial Intelligence) ∃ evidence that the patient is not suffering from pericarditis with 0. 3 m 2(x)= 0. 3 0. 7 if x= Ω if x= {heart attack, pulmonary embolism, aortic dissection} 0 elsewhere Combination of m 1 and m 2

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. A I (Artificial Intelligence) 0. 18 if x= Ω 0. 28 if x= {heart attack} m 1 m 2 = 0. 12 if x= {heart attack, pericarditis} 0. 42 if x= {heart attack, pulmonary embolism, ad} 0 otherwise (P. 242 -245 Rich) Ω = { Cold, Flu, Headache, Meningitis} Fever supporting {C, F, M} with 0. 6 of belief. 患者가 熱이 있으면 {C, F, M}일 確率이 0. 6이라고 假定 m 1({C, F, M}) = 0. 6 m 1(Ω) = 0. 4 Extreme nausea { C, F, H} : 0. 6 m 2({C, F, H}) = 0. 7 m 2(Ω) = 0. 3